Question
What are some examples to illustrate when data in a table does not represent a proportional relationship?
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GPT-4o mini
A proportional relationship implies that two quantities increase or decrease in a consistent ratio. In contrast, data in a table may not represent a proportional relationship in several scenarios. Here are some examples to illustrate this:
### Example 1: Fixed Increase
| x | y |
|---|---|
| 1 | 2 |
| 2 | 5 |
| 3 | 8 |
| 4 | 11|
In this table, as \(x\) increases by 1, \(y\) increases by 3. The ratio \(y/x\) is not constant (2, 2.5, 2.67, 2.75), so this does not represent a proportional relationship.
### Example 2: Non-linear Relationship
| x | y |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
Here, \(y\) represents \(x^2\). The ratios \(y/x\) do not remain consistent (1, 2, 3, 4), thus indicating that the relationship is quadratic rather than proportional.
### Example 3: Inversely Related Quantities
| x | y |
|---|---|
| 1 | 10 |
| 2 | 5 |
| 5 | 2 |
| 10 | 1 |
In this case, as \(x\) increases, \(y\) decreases. The ratios \(y/x\) (10, 2.5, 0.4, 0.1) vary widely, showing that the relationship is not proportional but rather inversely proportional.
### Example 4: Constant Values
| x | y |
|---|---|
| 1 | 5 |
| 2 | 5 |
| 3 | 5 |
| 4 | 5 |
In this table, \(y\) remains constant while \(x\) increases. Here, the ratio \(y/x\) changes (5, 2.5, 1.67, 1.25), which means it's not proportional.
### Example 5: Mixed Increments
| x | y |
|---|---|
| 1 | 1 |
| 2 | 3 |
| 3 | 7 |
| 4 | 12 |
In this case, the increments of \(y\) are inconsistent (the differences between values are 2, 4, and 5), resulting in varying ratios \(y/x\) (1, 1.5, 2.33, 3) that show no consistent proportional relationship.
### Conclusion
In each of these examples, the absence of a constant ratio between \(x\) and \(y\ or other forms of consistency demonstrates that the relationships do not adhere to proportionality. Instead, they exhibit various types of mathematical relationships, including linear non-proportionality, quadratic relationships, constant values, and inversely related quantities.
### Example 1: Fixed Increase
| x | y |
|---|---|
| 1 | 2 |
| 2 | 5 |
| 3 | 8 |
| 4 | 11|
In this table, as \(x\) increases by 1, \(y\) increases by 3. The ratio \(y/x\) is not constant (2, 2.5, 2.67, 2.75), so this does not represent a proportional relationship.
### Example 2: Non-linear Relationship
| x | y |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
Here, \(y\) represents \(x^2\). The ratios \(y/x\) do not remain consistent (1, 2, 3, 4), thus indicating that the relationship is quadratic rather than proportional.
### Example 3: Inversely Related Quantities
| x | y |
|---|---|
| 1 | 10 |
| 2 | 5 |
| 5 | 2 |
| 10 | 1 |
In this case, as \(x\) increases, \(y\) decreases. The ratios \(y/x\) (10, 2.5, 0.4, 0.1) vary widely, showing that the relationship is not proportional but rather inversely proportional.
### Example 4: Constant Values
| x | y |
|---|---|
| 1 | 5 |
| 2 | 5 |
| 3 | 5 |
| 4 | 5 |
In this table, \(y\) remains constant while \(x\) increases. Here, the ratio \(y/x\) changes (5, 2.5, 1.67, 1.25), which means it's not proportional.
### Example 5: Mixed Increments
| x | y |
|---|---|
| 1 | 1 |
| 2 | 3 |
| 3 | 7 |
| 4 | 12 |
In this case, the increments of \(y\) are inconsistent (the differences between values are 2, 4, and 5), resulting in varying ratios \(y/x\) (1, 1.5, 2.33, 3) that show no consistent proportional relationship.
### Conclusion
In each of these examples, the absence of a constant ratio between \(x\) and \(y\ or other forms of consistency demonstrates that the relationships do not adhere to proportionality. Instead, they exhibit various types of mathematical relationships, including linear non-proportionality, quadratic relationships, constant values, and inversely related quantities.